A188404 Number of (3*n) X n binary arrays with rows in nonincreasing order, 3 ones in every column and no more than 3 ones in any row.
1, 4, 23, 214, 2698, 44288, 902962, 22262244, 648446612, 21940389584, 849992734124, 37273085398456, 1831837147680872, 100066601315825216, 6031974947471801512, 398733149802770699792, 28744536471179273843088, 2248840133521868856571456, 190105368229118222009348848
Offset: 1
Keywords
Examples
All solutions for 6 X 2: ..1..1....1..0....1..1....1..1 ..1..1....1..0....1..0....1..1 ..1..0....1..0....1..0....1..1 ..0..1....0..1....0..1....0..0 ..0..0....0..1....0..1....0..0 ..0..0....0..1....0..0....0..0
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..320
- Frédéric Chyzak, Marni Mishna, Bruno Salvy, Effective Scalar Products for D-finite Symmetric Functions, arXiv:math/0310132v3 (version 2012)
- P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, pp. 584-585.
- Goulden, I. P. and Jackson, D. M., Labelled graphs with small vertex degrees and $P$-recursiveness, SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60-66. MR0819706 (87k:05093).
Programs
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Mathematica
max=20; f[x_]:=Sum[a[n]*(x^(n)/n!),{n,0,max}]; a[0]=1; a[1]=1; coef = CoefficientList[9*x^3*(x^4 - x^2 + x-2)*f''[x] - 3*(x^10 - 2*x^8 + 2*x^6 - 6*x^5 + 8*x^4 + 2*x^3 + 8*x^2 + 16*x - 8)*f'[x] + (x^11 + x^10 - 6*x^9 - 4*x^8 + 11*x^7 - 15*x^6 + 8*x^5 - 2*x^3 + 12*x^2 - 24*x - 24)*f[x],x]; Table[a[n],{n,0,max}]/.Solve[Thread[coef[[2;;max]]==0]][[1]]//Rest (* Vaclav Kotesovec, Sep 14 2014 *) Flatten[{1,RecurrenceTable[{-(-7+n) * (-6+n) * (-5+n) * (-4+n) * (-3+n) * (-2+n) * (-1+n) * (-7+3 * n) * (4+114 * n-144 * n^2+27 * n^3) * a[-8+n]-(-6+n) * (-5+n) * (-4+n) * (-3+n) * (-2+n) * (-1+n) * (2+3 * n) * (-281+483 * n-225 * n^2+27 * n^3) * a[-7+n]+(-5+n) * (-4+n) * (-3+n) * (-2+n) * (-1+n) * (85-60 * n+9 * n^2) * (4+114 * n-144 * n^2+27 * n^3) * a[-6+n]+4 * (-4+n) * (-3+n) * (-2+n) * (-1+n) * (1112-3117 * n+2781 * n^2-864 * n^3+81 * n^4) * a[-5+n]-(-3+n) * (-2+n) * (-1+n) * (1820+4458 * n-14454 * n^2+10395 * n^3-2754 * n^4+243 * n^5) * a[-4+n]-3 * (-2+n) * (-1+n) * (-1892+6068 * n-7239 * n^2+3915 * n^3-945 * n^4+81 * n^5) * a[-3+n]-9 * (-1+n)^2 * (296+4904 * n-8256 * n^2+4563 * n^3-1026 * n^4+81 * n^5) * a[-2+n]-6 * (-728+9186 * n-16911 * n^2+10989 * n^3-2835 * n^4+243 * n^5) * a[-1+n]+12 * (-10+3 * n) * (-281+483 * n-225 * n^2+27 * n^3) * a[n]==0,a[2]==4,a[3]==23,a[4]==214,a[5]==2698,a[6]==44288,a[7]==902962,a[8]==22262244,a[9]==648446612},a,{n,2,20}]}] (* Vaclav Kotesovec, Sep 15 2014 *)
Formula
See Goulden and Jackson for the e.g.f. - N. J. A. Sloane, Sep 02 2013
Recurrence (for n>9): 12*(3*n - 10)*(27*n^3 - 225*n^2 + 483*n - 281)*a(n) = 6*(243*n^5 - 2835*n^4 + 10989*n^3 - 16911*n^2 + 9186*n - 728)*a(n-1) + 9*(n-1)^2*(81*n^5 - 1026*n^4 + 4563*n^3 - 8256*n^2 + 4904*n + 296)*a(n-2) + 3*(n-2)*(n-1)*(81*n^5 - 945*n^4 + 3915*n^3 - 7239*n^2 + 6068*n - 1892)*a(n-3) + (n-3)*(n-2)*(n-1)*(243*n^5 - 2754*n^4 + 10395*n^3 - 14454*n^2 + 4458*n + 1820)*a(n-4) - 4*(n-4)*(n-3)*(n-2)*(n-1)*(81*n^4 - 864*n^3 + 2781*n^2 - 3117*n + 1112)*a(n-5) - (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(9*n^2 - 60*n + 85)*(27*n^3 - 144*n^2 + 114*n + 4)*a(n-6) + (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(3*n + 2)*(27*n^3 - 225*n^2 + 483*n - 281)*a(n-7) + (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(3*n - 7)*(27*n^3 - 144*n^2 + 114*n + 4)*a(n-8). - Vaclav Kotesovec, Sep 14 2014
Asymptotics (Chyzak, 2003): a(n) ~ c * (n!)^(3/2) * (sqrt(3)/2)^n * exp(sqrt(3*n)) / n^(3/4), where c = 1/sqrt(2) * exp(3/4) / (2*Pi)^(3/4) = 0.37719937314536... . - Vaclav Kotesovec, Sep 14 2014
Extensions
More terms from Vaclav Kotesovec, Sep 14 2014
Comments